Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2201.01662

Path group actions induced by sigma-actions and affine Kac-Moody symmetric spaces of group type

Masahiro Morimoto
View original source ↗Read source document ↗Export metadata ↓

Abstract

In 1995, C.-L. Terng associated to each hyperpolar action on a compact symmetric space, a hyperpolar proper Fredholm (PF) action on a Hilbert space. This is a group action by an infinite dimensional path group and it acts on a Hilbert space via the gauge transformations. Those two hyperpolar actions are related through an equivariant Riemannian submersion called the parallel transport map and they have close relations to the infinite dimensional symmetric spaces called affine Kac-Moody symmetric spaces. In this paper we define a linear isomorphism between Hilbert spaces and show that it is equivariant with respect to the gauge transformations and is compatible with the parallel transport map. Using this isomorphism we extend and unify all known computational results of principal curvatures of PF submanifolds in Hilbert spaces. Especially we study the submanifold geometry of orbits of hyperpolar PF actions associated to sigma-actions and give new examples of austere PF submanifolds in Hilbert spaces. Moreover we show that the isomorphism between Hilbert spaces given here corresponds to a natural isomorphism between affine Kac-Moody symmetric spaces of group type.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Masahiro Morimoto. 2023-03-01. Path group actions induced by sigma-actions and affine Kac-Moody symmetric spaces of group type. https://arxiv.org/abs/2201.01662

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Ancient mean curvature flow asymptotic to a minimal quadratic cone

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an $O(n)\times O(m)$ symmetric minimal quadratic cone for $n +m \geq 10$, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a $\textbf{singular minimal cone as the asymptotic model}$.

math.DG↗
arXiv2026

Log-Concavity of First Dirichlet Eigenfunctions on $\mathbb{CP}^2$

We study the log-concavity property of first Dirichlet eigenfunctions on domains in $\mathbb{CP}^2$. For every smooth $1$-convex domain $Ω\subset\mathbb{CP}^2$, we prove the quantitative estimate \[ \nabla^2(-\log u) > \max\left\{ψ(s),\frac85\right\}g, \text{ where } ψ(|\grad f|^2) = \frac{s}{\sqrt{1+s}}-\log(1+s), \] for its first Dirichlet eigenfunction $u$. In particular, $u$ is strictly log-concave. As consequences, we obtain a uniform convexity estimate for the regular level sets of $u$ and the fundamental gap bound $λ_2-λ_1>46/5$.

math.DG↗
arXiv2026

Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications

We prove a Pólya--Szegö inequality for functions defined on an $n$-dimensional submanifold $Σ$ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on $\mathbb R^n$, and the constant depends on the $L^n$-norm of the mean curvature of $Σ$ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo--Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser--Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya--Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.

math.DG↗